# New differential operator and non-collapsed $RCD$ spaces

## TL;DR

This paper characterizes non-collapsed compact RCD(K, N) spaces, confirming a conjecture and introducing a new differential operator derived from heat kernel embeddings, pioneering the use of geometric flow in RCD space analysis.

## Contribution

It provides explicit formulas for the Laplacian on RCD spaces and confirms a key conjecture, advancing the understanding of non-collapsed RCD spaces.

## Key findings

- Confirmed a conjecture relating weakly non-collapsed and non-collapsed RCD spaces.
- Derived explicit formulas for the Laplacian via heat kernel embeddings.
- Applied geometric flow techniques to the study of RCD spaces.

## Abstract

We show characterizations of non-collapsed compact $RCD(K, N)$ spaces, which in particular confirm a conjecture of De Philippis-Gigli on the implication from the weakly non-collapsed condition to the non-collapsed one in the compact case. The key idea is to give the explicit formula of the Laplacian associated to the pull-back Riemannian metric by embedding in $L^2$ via the heat kernel. This seems the first application of geometric flow to the study of RCD spaces.